Finite Group Databasebeta

← #10818120

Subgroups — #10818120

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79210C211yesmaximal, characteristic, complemented
2#78947C211yesmaximal, characteristic, complemented
3#78774C211yesmaximal, characteristic, complemented
4C2×#4754C211yesmaximal, characteristic
5#78560C211yesmaximal, characteristic
6C3×#1930C211yesmaximal, characteristic, complemented
7#70674C211yesmaximal, characteristic, complemented
8#70348C211yesmaximal, characteristic, complemented
9#70260C211yesmaximal, characteristic, complemented
10#70102C211yesmaximal, characteristic, complemented
11#69890C211yesmaximal, characteristic, complemented
12#69005C211yesmaximal, characteristic
13#68985C211yesmaximal, characteristic, complemented
14#68931C211yesmaximal, characteristic, complemented
15#68927C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17#5812C2211yescharacteristic
18C2×#942C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#908C2211yescharacteristic
21C2×#907C2211yescharacteristic
22#5523C2211yescharacteristic
23#5521C2211yescharacteristic
24#5504C2211yescharacteristic
25C3×#428C2211yescharacteristic
26C3×#419C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#101C2211yescharacteristic
29C3×#405C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#825C2211yescharacteristic
32#5131C2211yescharacteristic
33C2×#823C2211yescharacteristic
34C2×#822C2211yescharacteristic
35#5122C2211yescharacteristic
36#5121C2211yescharacteristic
37#5121C2211yescharacteristic
38#5113C2211yescharacteristic
39C2×#821C2211yescharacteristic
40#5101C2211yescharacteristic
41C2×#815C2211yescharacteristic
42#4998C2211yescharacteristic
43#4988C2211yescharacteristic
44#4964C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4881—21no—
47D3×#107—21no—
48#4861—21no—
49#4860—21no—
50#4817—21no—
51#4817—21no—
52#4817—21no—
53#4790—21no—
54#4789—21no—
55#4788—21no—
56#4788—21no—
57#4788—21no—
58#4754C418yescomplemented
59#4711—21no—
60#4710—21no—
61C4×D24—21no—
62C4×#204—21no—
63C3×#343C2211yescharacteristic
64C3×#341C2211yescharacteristic
65#4555C2211yescharacteristic
66#4555C2211yescharacteristic
67#4554C2211yescharacteristic
68#4532C2211yescharacteristic
69C2×#471—31no—
70#2949—31no—
71#2947—31no—
72#2946—31no—
73#2913—31no—
74#1973—31no—
75#1952—31no—
76#1931—31no—
77#1930D311yescharacteristic, complemented
78#1883—31no—
79#1882—31no—
80#1857—31no—
81#1856—31no—
82#1773—31no—
83#1765—31no—
84C22×C3×Q8C2311yescharacteristic
85C2×#235C2311yescharacteristic
86C22×D12C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#117C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93Q8×Dic3—21no—
94Q8×Dic3—21no—
95Q8×Dic3—21no—
96Q8×Dic3—21no—
97C2×#216—21no—
98#942—22no—
99#942C2×C414yescomplemented
100C2×#215—21no—
101C2×#215—21no—
102C2×#215—21no—
103C2×#215—22no—
104C2×#215C2×C414yescomplemented
105C4×#67—41no—
106C2×(C2×D3)⋊C4—41no—
107#926—41no—
108#926C2311yescharacteristic
109#926C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#908—22no—
115#908C2×C414yes—
116#907—22no—
117#907C2×C414yescomplemented
118C2×D24—21no—
119C2×#204—21no—
120C2×ΓL(1,𝔽25)—21no—
121C2×D3×C8—21no—
122#893—21no—
123#893—21no—
124#893—21no—
125#892—21no—
126#892—21no—
127D3×C4⋊C4—21no—
128D3×C4⋊C4—21no—
129#887—21no—
130#67⋊C4—41no—
131D3×C22⋊C4—41no—
132C4×D12—21no—
133C4×D12—21no—
134C4×D12—21no—
135C4×Dic6—21no—
136C3×#107—21no—
137C3×#106—21no—
138C3×#103—21no—
139C3×#103—21no—
140C3×#103—21no—
141C3×#103—21no—
142C3×#101C2×C414yescomplemented
143C3×#97—21no—
144C3×C4×C8—21no—
145C3⋊#95—41no—
146#825—22no—
147#823C2×C414yescomplemented
148#822C2×C414yescomplemented
149#821—21no—
150#821—21no—
151#821—21no—
152#821—21no—
153#821D411yescharacteristic
154#821D411yescharacteristic
155#818—21no—
156#818—21no—
157#817—21no—
158#817—21no—
159#816D411yescharacteristic
160#816D411yescharacteristic
161#815—21no—
162#815—21no—
163#815—22no—
164C8×Dic3—21no—
165C8×Dic3—21no—
166#808—21no—
167#808D411yescharacteristic
168#808D411yescharacteristic
169#807—21no—
170#807—21no—
171#807—21no—
172#807D411yescharacteristic
173#807D411yescharacteristic
174C2×#137—31no—
175C2×AGL(1,ℤ/8)—31no—
176C22×SD16—31no—
177C2×#131—31no—
178#520—31no—
179#520—31no—
180#519—31no—
181#518—31no—
182#471—38no—
183#442—61no—
184#441—61no—
185#440—61no—
186#440—61no—
187#440—61no—
188#440—61no—
189C4×Q16—61no—
190C4×SD16—61no—
191C4×SD16—61no—
192C4×SD16—61no—
193C4×SD16—61no—
194C4×D8—61no—
195C2×#108—31no—
196#428—31no—
197#428—31no—
198#428—31no—
199#428—61no—
200#428—61no—
201#428C2×D311yescharacteristic
202#427—31no—
203#427—61no—
204C2×#107—31no—
205C2×#107—61no—
206C2×#104—31no—
207#419C2×D311yescharacteristic
208#418—31no—
209#416—31no—
210C2×#103C2×D311yescharacteristic
211C2×#102—31no—
212C2×#101C2×D311yescharacteristic
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405C2×D311yescharacteristic
217C2×#97—31no—
218#391—31no—
219#390—31no—
220#343—31no—
221#343C2×D311yescharacteristic
222#341C2×D311yescharacteristic
223#339—31no—
224#339—31no—
225#339—31no—
226C23×D3—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—21no—
231C2×C3×Q8—21no—
232C2×C3×Q8C22×C412yes—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4C2411yescharacteristic
236C2×#67—21no—
237C22×Dic3—21no—
238C22×Dic3—21no—
239#235—21no—
240#235—21no—
241#235C22×C412yes—
242#235#4112yes—
243C2×D12—21no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12—21no—
247C2×D12—21no—
248C2×D12C22×C412yes—
249C2×D12C2×D411yescharacteristic
250C2×C4×D3—21no—
251C2×C4×D3—41no—
252C2×C4×D3—41no—
253C2×C4×D3—41no—
254C2×C4×D3—41no—
255C2×Dic6C2×D411yescharacteristic
256C3×M16—21no—
257C3×M16—21no—
258C3×M16—21no—
259C3×M16—21no—
260C3×M16C22×C412yes—
261C3×M16C22×C412yes—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C2×C3×C8—21no—
267C2×C3×C8C2×D411yescharacteristic
268C2×C3×C8C2×D411yescharacteristic
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4C2×D411yescharacteristic
272C3×C4⋊C4C2×D411yescharacteristic
273C3×C22⋊C4—21no—
274C3×C42—21no—
275#217—21no—
276#217—21no—
277#216—24no—
278#215—24no—
279#215—24no—
280#215—24no—
281(C2×D3)⋊C4—41no—
282(C2×D3)⋊C4—41no—
283(C2×D3)⋊C4—41no—
284(C2×D3)⋊C4—41no—
285(C2×D3)⋊C4—82no—
286C4⋊Dic3—21no—
287C4⋊Dic3—21no—
288C4⋊Dic3—41no—
289C4⋊Dic3—41no—
290C4⋊Dic3C2×D411yescharacteristic
291C4⋊Dic3C2×D411yescharacteristic
292C4⋊Dic3C2×D411yescharacteristic
293C4⋊Dic3C2×D411yescharacteristic
294Dic3⋊C4—41no—
295Dic3⋊C4—41no—
296C4×Dic3—21no—
297C4×Dic3—21no—
298C4×Dic3—21no—
299C4×Dic3—21no—
300C4×Dic3—41no—
301C4×Dic3—41no—
302C4×Dic3—41no—
303C4×Dic3—41no—
304#208—21no—
305#208C22×C412yes—
306C2×C3⋊C8—21no—
307C2×C3⋊C8C22×C412yes—
308C2×C3⋊C8C2×D411yescharacteristic
309C2×C3⋊C8C2×D411yescharacteristic
310C2×C3⋊C8#4112yes—
311D24—24no—
312#204—24no—
313ΓL(1,𝔽25)—24no—
314D3×C8—24no—
315C2×#41—31no—
316C22×Q8C22×D311yescharacteristic
317C22×D4—31no—
318#137—34no—
319#137—62no—
320AGL(1,ℤ/8)—34no—
321AGL(1,ℤ/8)—62no—
322C2×Q16—61no—
323C2×SD16—34no—
324C2×SD16—61no—
325C2×SD16—62no—
326C2×SD16—61no—
327C2×SD16—61no—
328C2×SD16—61no—
329C2×D8—61no—
330#131—34no—
331#131—62no—
332C2×M16—31no—
333C2×M16—61no—
334C2×M16C22×D311yescharacteristic
335C2×M16C22×D311yescharacteristic
336C2×M16C22×D311yescharacteristic
337C22×C8—31no—
338C22×C8—61no—
339C22×C8C22×D311yescharacteristic
340C4×Q8—61no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×D4—61no—
347C4×D4—61no—
348C4×D4—61no—
349C4×D4—61no—
350C4×D4—61no—
351C4×D4—61no—
352C4×D4—121no—
353C4×D4—121no—
354#117—31no—
355#117—31no—
356#117—61no—
357#117—61no—
358#117—121no—
359#117C22×D311yescharacteristic
360C2×C4⋊C4—31no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—121no—
365C2×C22⋊C4—121no—
366C2×C42—31no—
367#108—34no—
368#107—21no—
369#107—31no—
370#107—31no—
371#107—61no—
372#107—61no—
373#107—61no—
374#107—61no—
375#107—61no—
376#106—21no—
377#106—31no—
378#106—31no—
379#106—61no—
380#106—61no—
381#106—61no—
382#104—62no—
383#103—21no—
384#103—21no—
385#103—21no—
386#103—21no—
387#103—31no—
388#103—31no—
389#102—31no—
390#102—31no—
391#102—61no—
392#102—61no—
393#102—61no—
394#102—61no—
395#101C4×D314yescomplemented
396#100—34no—
397#97—21no—
398#97—62no—
399#97—61no—
400#97—61no—
401C4×C8—21no—
402C4×C8—61no—
403C4×C8—61no—
404#95—121no—
405C23×C3H(5,𝔽2)11yescharacteristic
406C22×D3—21no—
407C22×D3—22no—
408C22×D3—41no—
409C22×D3—41no—
410C22×D3—81no—
411C22×D3—81no—
412C3×Q8—22no—
413C3×Q8—22no—
414C3×Q8—41no—
415C3×Q8—41no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—81no—
423C2×C3×C4C23×C411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C22×D411yescharacteristic
426C2×C3×C4C2×#4111yescharacteristic
427C2×C3×C4C2×#4111yescharacteristic
428C2×C3×C42−1+411yescharacteristic
429#67—22no—
430#67—22no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—81no—
441D12—22no—
442D12—22no—
443D12—21no—
444D12—41no—
445D12—41no—
446D12C4×D412yes—
447C4×D3—22no—
448C4×D3—22no—
449C4×D3—161no—
450C4×D3—161no—
451C4×D3—161no—
452C4×D3—161no—
453Dic6—21no—
454Dic6C4×D412yes—
455C3×C8—22no—
456C3×C8—22no—
457C3×C8—21no—
458C3×C8—21no—
459C3×C8—41no—
460C3⋊C8—22no—
461C3⋊C8C4×D412yes—
462C3⋊C8C4×D412yes—
463C24—61no—
464#41—32no—
465#41—32no—
466#41—61no—
467#41—61no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—31no—
474C2×Q8C2×C4×D312yes—
475C2×D4—32no—
476C2×D4—31no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—61no—
482C2×D4—61no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—121no—
489C22×C4—121no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4C23×D311yescharacteristic
493Q16—64no—
494SD16—64no—
495SD16—64no—
496SD16—64no—
497SD16—64no—
498D8—64no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—21no—
503M16—32no—
504M16—64no—
505M16—61no—
506M16C2×C4×D312yes—
507M16C2×C4×D312yes—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—32no—
514C2×C8—31no—
515C2×C8—31no—
516C2×C8—32no—
517C2×C8—64no—
518C2×C8—61no—
519C2×C8D3×D411yescharacteristic
520C2×C8D3×D411yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—31no—
524C4⋊C4—31no—
525C4⋊C4—31no—
526C4⋊C4—31no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—121no—
530C4⋊C4—121no—
531C4⋊C4—121no—
532C4⋊C4—121no—
533C4⋊C4D3×D411yescharacteristic
534C4⋊C4D3×D411yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—61no—
537C22⋊C4—61no—
538C22⋊C4—121no—
539C22⋊C4—121no—
540C22⋊C4—121no—
541C22⋊C4—121no—
542C22⋊C4—242no—
543C42—21no—
544C42—61no—
545C42—61no—
546C42—61no—
547C42—61no—
548C42—121no—
549C42—121no—
550C42—121no—
551C42—121no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3#52011yescharacteristic
555C22×C3D4211yescharacteristic
556C22×C3#55311yescharacteristic
557C2×D3—22no—
558C2×D3—41no—
559C2×D3—41no—
560C2×D3—42no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—82no—
564C2×D3—161no—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—81no—
569C3×C4C2×C4×D411yescharacteristic
570C3×C4C2×C4×D411yescharacteristic
571C3×C4#54711yescharacteristic
572C3×C4#54811yescharacteristic
573Dic3—22no—
574Dic3—41no—
575Dic3—41no—
576Dic3—41no—
577Dic3—41no—
578Dic3—81no—
579Dic3—81no—
580C23—61no—
581C23—62no—
582C23—121no—
583C23—121no—
584C23—241no—
585C23—241no—
586C23#100911yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—32no—
590Q8—41no—
591Q8—41no—
592Q8—61no—
593D4—32no—
594D4—62no—
595D4—62no—
596D4—61no—
597D4—62no—
598D4—62no—
599D4—121no—
600D4—121no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—61no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—81no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—241no—
620C2×C4—481no—
621C2×C4—481no—
622C2×C4—481no—
623C2×C4—481no—
624C2×C4C22×C4×D311yescharacteristic
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4D3×#4111yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4#101011yescharacteristic
630C8—21no—
631C8—21no—
632C8—22no—
633C8—22no—
634C8—32no—
635C8—32no—
636C8—41no—
637C8—62no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3#236111yescharacteristic
641C2×C3#328811yescharacteristic
642C2×C3#328911yescharacteristic
643D3—42no—
644D3—82no—
645C22—21no—
646C22—21no—
647C22—62no—
648C22—121no—
649C22—121no—
650C22—122no—
651C22—242no—
652C22—241no—
653C22—241no—
654C22—481no—
655C22#552311yescharacteristic
656C22#560911yescharacteristic
657C22#570911yescharacteristic
658C4—41no—
659C4—41no—
660C4—42no—
661C4—62no—
662C4—81no—
663C4—121no—
664C4—121no—
665C4—121no—
666C4—121no—
667C4—241no—
668C4—241no—
669C4C4×D3×D411yescharacteristic
670C4C4×D3×D411yescharacteristic
671C4#565511yescharacteristic
672C4#567711yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—122no—
677C2—242no—
678C2#7547711yescharacteristic
679C2#8093211yescharacteristic
680C2#8094511yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1930C2
socleC22×C3D42
prime radicalC2×#908C22
cosocleC22×C3×C4C24
hypercenter#1930D3
hypocenterC3#23049
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1930

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #79210C2semidirect
2. #78947C2semidirect
3. #78774C2semidirect
6. C3×#1930C2semidirect
7. #70674C2semidirect
8. #70348C2semidirect
9. #70260C2semidirect
10. #70102C2semidirect
11. #69890C2semidirect
13. #68985C2semidirect
14. #68931C2semidirect
15. #68927C2semidirect
58. #4754C4semidirect
77. #1930D3semidirect
99. #942C2×C4semidirect
104. C2×#215C2×C4semidirect
117. #907C2×C4semidirect
142. C3×#101C2×C4semidirect
147. #823C2×C4semidirect
148. #822C2×C4semidirect
395. #101C4×D3semidirect
673. C3#23049semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#792102
#789472
#787742
C2×#47542
#785602
C3×#19302
#706742
#703482
#702602
#701022
#698902
#690052
#689852
#689312
#689272
#230493